# Error bound for hybrid models of two

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Error bound for hybrid models of two

Error bound for hybrid models of two-scaled stochastic reaction systems Tobias Jahnke and Vikram Sunkara 1 Introduction Biological systems such as gene-regulatory networks and cell metabolic processes consist of multiple species which are undergoing transformations via a set of reaction channels. If all populations are sufficiently large, then the evolution of the concentrations over time can be modeled by the classical reaction-rate equation, i.e. a system of ordinary differential equations; cf. [23]. In many applications, however, some of the species occur in low amounts, and it was observed that small stochastic fluctuations in their populations can cascade large effects to the other species. Important examples are gene-regulatory networks where the evolution of the entire system depends crucially on the stochastic behavior of a rather small number of transcription factors. In order to capture these effects, such systems must be described by a Markov jump processes, which respects the inherent discrete nature of the system and its stochastic interactions. The associated time-dependent probability distribution is the solution of the Chemical Master Equation (CME), but solving the CME is a considerable challenge, as the size of the state space scales exponentially in the number of species (curse of dimensionality). For this reason, Monte Carlo approaches based on the stochastic simulation algorithm from [6] or related methods are often used. In an alternative line of research, numerical techniques have been applied to the CME in order to reduce the number of degrees of freedom, e.g. optimal state space truncation [1, 26, 27], spectral approximation [4], adaptive wavelet compression [16, 20], sparse grids [11], or tensor product approximation [2, 10, 18, 22, 21] among othTobias Jahnke Karlsruhe Institute of Technology, Department of Mathematics, Kaiserstr. 93, D-76133 Karlsruhe, Germany, e-mail: [email protected] Vikram Sunkara Karlsruhe Institute of Technology, Department of Mathematics, Kaiserstr. 93, D-76133 Karlsruhe, Germany, e-mail: [email protected] 1 2 Tobias Jahnke and Vikram Sunkara ers. But in spite of the progress achieved with these approaches, many biological systems are still out of reach of direct numerical approximation. The size of the problem can be significantly reduced if only species with low populations are described by a probability distribution, whereas the abundant species are represented by (conditional) moments. This approach is motivated by the famous result in [23] which states, roughly speaking, that stochastic fluctuations in large populations are insignificant. In the last years, this has inspired the development of hybrid models where a low-dimensional CME is coupled to ordinary differential equations similar to the classical reaction-rate equation; cf. [5, 9, 11, 12, 13, 17, 25, 28]. In this article, we analyze the accuracy of a hybrid model called MRCE (model reduction based on conditional expectations). This approach has been proposed in [9, 17, 25], and it was demonstrated numerically that MRCE captures the critical bi-modal solution profiles which appear in certain applications. In [9, 17, 28], numerical techniques for MRCE were introduced, and an error bound for the modeling error was proven in [28]. In the present article, we make the additional assumption that the reaction system involves two scales, i.e. that the ratio between the small and large populations is proportional to a scaling parameter 0 < ε 1. For such two-scaled systems, we prove that the modeling error of the MRCE approximation is proportional to ε. The proof blends ideas and techniques from [19] and [28]. 2 The Chemical Master Equation of two-scale reaction systems We consider a partitioned reaction system with two groups of species denoted by S1 , . . . , Sd and Sd+1 , . . . , Sd+D , respectively, with d, D ∈ N. Let X(t) ∈ Nd0 be the vector whose entries X1 (t), . . . Xd (t) indicate how many copies of each of the species S1 , . . . , Sd exist at time t ∈ [0,tend ], and let Y (t) = (Y1 (t), . . . ,YD (t)) contain the copy numbers of Sd+1 , . . . , Sd+D . The species interact via r ∈ N reaction channels , R1 , . . . , Rr , each of which is represented by a scheme d Rj : D cj ∑ a jk Sk + ∑ b jk Sd+k −→ k=1 k=1 d D ∑ abjk Sk + ∑ bb jk Sd+k , k=1 (1) k=1 with a jk , abjk , b jk , b b jk ∈ N0 and c j > 0. If the j-th reaction channel fires, then the population numbers jump from the current state (X(t),Y (t)) = (n, m) ∈ Nd0 × ND 0 to the new state (n, m) + (ν j , µ j ), where (ν j , µ j ) ∈ Zd+D is the stoichiometric vector associated to R j , i.e. T ν j = abj1 − a j1 , . . . , abjd − a jd ∈ Zd T µj = b b j1 − b j1 , . . . , b b jd − b jd ∈ ZD . Error bound for hybrid models of two-scaled stochastic reaction systems 3 In stochastic reaction kinetics, the function t 7→ (X(t),Y (t)) is a realization of a Markov jump process; cf. [6, 14]. According to [6] the transition rates of this process depend on the propensity functions of the reaction channels. We assume that the propensity function of R j has the form α j (n)β j (m) with d α j (n) = c j ∏ k=1 nk , a jk β j (m) = ε γ( j)−1 |b j | ε D ∏ k=1 mk , b jk (2) where |b j | = ∑di=1 b ji , and where 0 < ε 1 is a scaling parameter discussed below. The value of γ depends on whether or not the population numbers of the first group of species change when R j fires. To be more precise, we partition the index set {1, . . . , r} into J1 = {1, . . . , r} \ J0 J0 = j ∈ {1, . . . , r} : ν j = (0, . . . , 0)T , and let γ be the indicator function γ( j) = 0 if j ∈ J0 , 1 if j ∈ J1 . (3) The reason for this particular scaling is the following: if (X(t),Y (t)) = (n, m) ∈ Nd0 × −1 , then α (n)β (m) = O(c ε γ( j)−1 ) for all j = ND j j j 0 with n ∈ O(1) and m ∈ O ε 1, . . . , r. Hence, the population numbers of S , . . . , S may change with a rate d+1 d+D of O ε −1 , whereas the populations of S1 , . . . , Sd only change with a rate of O(1), provided that c j = O(1) for all j. For initial data X(0) = O(1) and Y (0) = O ε −1 , one can thus expect that E(X(t)) = O(1) and E(Y (t)) = O ε −1 on bounded time intervals. Hence, ε is roughly speaking the ratio between the small and the large population numbers of the two groups S1 , . . . , Sd and Sd+1 , . . . , Sd+D , respectively. This scaling was extensively motivated and illustrated in [19], and a very similar scaling was considered in [25]. For d = 0 and α j (n) = c j , our scaling coincides with the thermodynamic limit which has been analyzed in [23]. For ε = 1, there is no qualitative difference between the two groups of species, such that our setting corresponds to the situation in [6] where no partition nor scaling is considered. Let p(t, n, m) be the probability that at time t ≥ 0 the process is in state (n, m) ∈ Nd0 × ND 0 , i.e. the probability that there are nk copies of Sk for k = 1, . . . , d, and ml copies of Sd+l for l = 1, . . . , D. It is well-known (see [6, 7]) that the probability distribution p evolves according to the Chemical Master Equation (CME) r ∂t p(t, n, m) = (4) ∑ α j (n − ν j )β j (m − µ j )p(t, n − ν j , m − µ j ) j=1 − α j (n)β j (m)p(t, n, m) ∀(n, m) ∈ Nd0 × ND 0 p(0, n, m) = p0 (n, m) (5) 4 Tobias Jahnke and Vikram Sunkara with the convention that p(t, n − ν j , m − µ j ) = 0 if n − ν j 6∈ Nd0 or m − µ j 6∈ ND 0 . For the sake of a more compact notation we define the shift operators S1j and S2j by S1j u(n, m) = S2j u(n, m) = u(n − ν j , m) 0 if n − ν j ∈ Nd0 else u(n, m − µ j ) 0 if m − µ j ∈ ND 0 else for u : Nd0 × ND 0 −→ R. The two shift operators commute, i.e. u(n − ν j , m − µ j ) if n − ν j ∈ Nd0 , m − µ j ∈ ND 1 2 2 1 0 S j S j u(n, m) = S j S j u(n, m) = 0 else. Products of functions are to be understood entry-wise, and applying a shift operator to a product u(n, m)v(n, m) is to be understood in the sense that S1j uv (n, m) = S1j (uv) (n, m) = u(n − ν j , m)v(n − ν j , m) = S1j u S1j v (n, m). With these operators, the CME (4) can be reformulated as r ∂t p = ∑ (S1j S2j − I) (α j β j p) . (6) j=1 The chemical master equation (6) is considered on the space n o `1 = u : Nd0 × ND 0 −→ R : ∑ ∑ |u(n, m)| < ∞ . n∈Nd0 m∈ND 0 of absolutely summable functions on Nd0 × ND 0 . This is a straightforward extension of the standard `1 -space. For vector-valued functions u = (u1 , . . . , uN ) : Nd0 ×ND 0 −→ RN with some N > 1, u ∈ `1 means that u j ∈ `1 for all j = 1, . . . , N. The space `1 is endowed with the norm kuk`1 = ∑ ∑ |u(n, m)| n∈Nd0 m∈ND 0 where | · | = | · |1 is the 1-norm on RN . We set X 0 = `1 and define the spaces X i via the recursion n o X i+1 = u ∈ X i | (n, m) 7→ mk u(n, m) ∈ X i for all k ∈ {1, . . . , D} . If p(t, ·, ·) ∈ `1 is the solution of the CME (6), then p1 (t, n) = ∑m p(t, n, m) is the marginal distribution of p(t, ·, ·), and if p1 (t, n) 6= 0, then p2 (t, m | n) = p(t, n, m) p1 (t, n) (7) Error bound for hybrid models of two-scaled stochastic reaction systems 5 is the conditional probability that at time t there are m j particles of S j given there are ni particles of Si (i ∈ {1, . . . , d}, j ∈ {d + 1, . . . , d + D}). If p(t, ·, ·) ∈ X 2 , then the conditional central moments ξ (t, n) = ∑ mp2 (t, m | n) (8) m∈ND 0 Cξ (t, n) = ∑D T m − ξ (t, n) m − ξ (t, n) p2 (t, m | n) m∈N0 exist provided that p1 (t, n) 6= 0. 3 Model reduction based on conditional expectations Solving the CME (4) or (6) numerically is a considerable challenge. First, the infinite state space Nd0 × ND 0 has to be truncated; this causes an error which has been analyzed in [26]. The truncated state space is finite, but still (d + D)-dimensional, and the total number of states is usually so large that standard numerical schemes cannot be applied. On the other hand, the solution of the CME often provides more information than actually needed to understand the biological process. In many applications, one is mainly interested in the question how the stochastic behavior of S1 , . . . , Sd affects the dynamics of Sd+1 , . . . , Sd+D . If the population numbers of Sd+1 , . . . , Sd+D are sufficiently large, then stochastic fluctuations within their populations can be neglected according to [23]. In this case, it is sufficient to compute the marginal distribution p1 (t, n) of the species S1 , . . . , Sd along with conditional moments which measure the abundance of Sd+1 , . . . , Sd+D . This has motivated the construction of hybrid models: Instead of trying to solve the high-dimensional CME and then extracting the relevant information from the solution p(t, n, m), one derives a reduced set of equations, namely a low-dimensional CME for the marginal distribution coupled with other ODEs; cf. [5, 9, 11, 12, 13, 17, 25, 28]. Hybrid models have the advantage that the huge number of unknowns is significantly reduced, which makes the problem computationally feasible. The price to pay is that hybrid models involve structurally more complicated differential equations than the (linear) CME, and that such a model reduction causes a modeling error in addition to the numerical error. The following hybrid model has been derived in [17]: ∂t w = ∑ (S1j − I) α j β j (φ )w =: A(φ )w (9) j∈J1 ∂t (φ w) = r α j β j (φ )φ w + ∑ µ j S1j α j β j (φ )w ∑ (S1j − I) j∈J1 =: F(φ , w) + G(φ , w) j=1 (10) 6 Tobias Jahnke and Vikram Sunkara For fixed φ , (9) is again a CME, but on the lower-dimensional state space Nd0 . The function w(t, n) approximates the marginal distribution p1 (t, n) of the full CME solution, whereas φ (t, n) approximates the conditional expectations ξ (t, n) defined in (8). This is why this model was called model reduction based on conditional expectations (MRCE) in [17]. It was demonstrated by numerical examples that MRCE captures certain bimodal solution profiles correctly, in contrast to simpler hybrid models proposed in the literature. Since w and φ do not depend on m any more, the (d + D)-dimensional state space of the CME is replaced by a d-dimensional state space, which reduces the computational costs considerably. Similar approaches have been proposed in [8, 9, 13, 15, 24, 25] for the CME and related differential equations. Approximating the conditional expectations ξ (t, n) has the drawback that (7) and hence (8) cannot be properly defined if p1 (t, n) = 0. The same applies to the approximations w(t, n) ≈ p1 (t, n) and φ (t, n) ≈ ξ (t, n). The hybrid model (9)-(10) is formulated in terms of w and φ w, but in order to evaluate the term β j (φ ) on the right-hand side, we have to divide φ (t, n)w(t, n) by w(t, n). This is only possible if w(t, n) > 0, and for w(t, n) ≈ 0 such a division causes numerical instability. Different strategies to cope with this problem have been proposed in [9, 13, 17, 25, 28]. Since the main goal of the present article is an analysis of the accuracy of MRCE, we will avoid such technical problems by the following assumption: Assumption 1 We assume that the CME (6) with initial condition (5) has a unique classical solution p(t, ·, ·) ∈ `1 with strictly positive marginal distribution p1 (t, ·), i.e. p1 (t, n) > 0 for all t ∈ [0,tend ] and all n ∈ Nd0 . This implies that p2 , ξ and Cξ are well-defined. Moreover, we assume that the hybrid model (9)-(10) with initial data w(0, n) = p1 (0, n) and φ (0, n) = ξ (0, n) (11) has a unique solution, and that w(t, ·) ∈ `1 is strictly positive for all t ∈ [0,tend ]. This assumption seems to be a strong simplification because in typical applications it can be observed that for every threshold parameter δ ∈ (0, 1), there are only finitely many states with p(t, n, m) ≥ δ . Roughly speaking, this means that p(t, n, m) ≈ 0 for “most of” the states. However, if p(t? , n? , m? ) = 0 for some t? > 0, then the state (n? , m? ) ∈ Nd0 × ND 0 cannot be reached from neither of the states which had nonzero probability at time t = 0. As a consequence, one could simply exclude (n? , m? ) from the state space to avoid the problem, and in this sense, Assumption 1 is not a severe restriction. Since numerical methods for solving (9)-(10) are not discussed in this article, numerical instabilities are not an issue here. 4 Error analysis for the hybrid model Since β j (m) defined in (2) depends on the scaling parameter ε and since β j (m) appears both in the CME (6) and in the hybrid model (9)-(10), the functions Error bound for hybrid models of two-scaled stochastic reaction systems 7 p(t, n, m), p1 (t, n), ξ (t, n), w(t, n), and φ (t, n) all depend1 on ε, too. In this section we prove that the modeling error of MRCE is bounded by Cε (see Theorem 1 below). Throughout the article, C denotes a generic constant which may have different values at different occurrences. The proof combines the arguments from [17, 19] with the analysis from [28] where systems with no scaling have been investigated. Our error analysis is based on the following assumptions. Assumption 2 For every j ∈ {1, . . . , r} we assume that |b j | ≤ 2. This is a natural assumption, because the probability of a trimolecular reaction is negligible according to [7, page 418]. Assumption 3 We assume that the solution of the CME (6) satisfies p(t, ·, ·) ∈ X 3 for t ∈ [0,tend ] and that (n, m) 7→ α j (n)p(t, n, m) ∈ X 3 for all j ∈ {1, . . . , r}. Assumption 4 We assume that sup sup |ξ (t, n)| ≤ t∈[0,tend ] n∈Nd 0 C , ε sup sup |Cξ (t, n)| ≤ t∈[0,tend ] n∈Nd 0 C , ε C sup sup |φ (t, n)| ≤ ε t∈[0,tend ] n∈Nd 0 with a constant which does not depend on ε. Moreover, we assume that all third central moments of p2 (t, · | n) are bounded by Cε −2 with a constant which does not depend on t ∈ [0,tend ], ε, and n ∈ Nd0 . Assumption 5 Suppose that there is a constant C > 0 such that for all t ∈ [0,tend ] and j ∈ {1, . . . , r} the bound max α j (·)u(t, ·)`1 ≤ C ku(t, ·)k`1 j=1,...,r holds for each of the following functions: u = p1 , u = β j ξ p1 − β j φ w, u = β j (ξ )ξ p1 − β j (φ )φ w. Note that Assumption 4 implies u ∈ `1 in each case. The following error bound for the modeling error of MRCE is the main result of this article. Theorem 1. Under the assumptions 1, 2, 3, 4, and 5, there is a constant Cb > 0 such that the approximation error of MRCE is bounded by 1 We do not make this dependency explicit in the notation in order to keep the equations as simple as possible. 8 Tobias Jahnke and Vikram Sunkara sup kp1 (t, ·) − w(t, ·)k`1 ≤ Cb ε (12) t∈[0,tend ] sup kξ (t, ·)p1 (t, ·) − φ (t, ·)w(t, ·)k`1 ≤ Cb . (13) t∈[0,tend ] If in addition |b j | ≤ 1 for all j ∈ J0 , |b j | = 0 for all j ∈ J1 , (14) then MRCE is even exact, i.e. we can choose Cb = 0 in (12) and (13). According to (13) the error of the approximation ξ p1 ≈ φ w remains bounded, but does not decrease when ε → 0. This is not obvious, because Assumption 4 implies that kξ (t, ·)p1 (t, ·)k`1 = O ε −1 and kφ (t, ·)w(t, ·)k`1 = O ε −1 . Multiplying both sides of (13) by ε shows that the relative error converges linearly in ε. Proof. It will be shown below in Lemma 2 and Lemma 3 that kp1 (t, ·) − w1 (t, ·)k`1 + ε kξ p1 (t, ·) − φ w(t, ·)k`1 ≤ Cb ε +C Zt ε k(ξ p1 − φ w) (s, ·)k`1 ds +C 0 Zt (15) kp1 (s, ·) − w(s, ·)k`1 ds. 0 for all t ∈ [0,tend ] with constants Cb and C which do not depend on t or ε. Hence, the Gronwall lemma yields kp1 (t, ·) − w1 (t, ·)k`1 + ε kξ p1 (t, ·) − φ w(t, ·)k`1 ≤ Cb ε which proves (12) and (13). Moreover, it will be shown that we can choose Cb = 0 in the special case (14). t u The remainder of this article is devoted to the proof of the Gronwall inequality (15). As a preparatory step, we prove the following lemma: Lemma 1. Let y : Nd0 −→ Rd , z : Nd0 −→ Rd with max |y(n)| ≤ C/ε, n∈Nd0 max |z(n)| ≤ C/ε, n∈Nd0 (16) and let u ∈ `1 and v ∈ `1 . Then for every j ∈ {1, . . . , r}, there is a constant C > 0 such that β j (y)u − β j (z)v 1 ≤ Cε γ( j) kyu − zvk 1 + ε −1 ku − vk 1 ` ` ` with γ( j) defined in (3). Note that the assumption (16) implies that yu − zv ∈ `1 . A similar lemma has been shown in [19, Lemma 4]. Proof. For |λ j | = 0 the assertion is obvious, because in this case β j (y) = ε γ( j)−1 is Error bound for hybrid models of two-scaled stochastic reaction systems 9 constant. If |λ j | = 1, then there is a k ∈ {1, . . . , d} such that β j (y) = ε γ( j) yk , and the assertion follows. If |λ j | = 2, then the propensity β j (y) takes the form c if k 6= l β j (y) = ĉ j ε γ( j)+1 yk yl with ĉ j = 1 j 2 c j if k = l for some k, l ∈ {1, . . . , d}. Thus, we have to bound the difference β j (y)u − β j (z)v = ĉ j ε γ( j)+1 (yk yl u − zk zl v) ! = ĉ j ε γ( j)+1 yk (yl u − zl v) + yk zl (v − u) + zl (yk u − zk v) . Since (16) implies that |εyk (n)| ≤ C and |εzl (n)| ≤ C, it follows that β j (y)u − β j (z)v 1 ≤ Cε γ( j) kyu − zvk 1 +Cε γ( j)−1 ku − vk 1 ` ` ` t u which proves the assertion. Lemma 2. Under the assumptions of Theorem 1 there are constants Cb ≥ 0 and C > 0 such that kp1 (t, ·) − w1 (t, ·)k`1 ≤ Cb ε + C Zt ε k(ξ p1 − φ w) (s, ·)k`1 ds 0 Zt +C kp1 (s, ·) − w(s, ·)k`1 ds. 0 for all t ∈ [0,tend ]. The constants Cb and C do not depend on t or ε, and we can choose Cb = 0 in the special case (14). Proof. From the definition of the marginal distribution p1 it follows that ∂t p1 = (S1j S2j − I)α j β j p + ∑ ∑ j∈J0 m∈ND (S1j S2j − I)α j β j p. ∑ ∑ j∈J1 m∈ND 0 0 The first sum vanishes, because S1j = I for j ∈ J0 , and ∑D (S2j − I)α j β j p = 0 (17) m∈N0 by Lemma 2 in [19]. Since S1j S2j − I = S1j (S2j − I) + (S1j − I) and since (17) implies ∑ ∑D S1j (S2j − I)α j β j p = 0, j∈J1 m∈N 0 we obtain 10 Tobias Jahnke and Vikram Sunkara ∂t p1 = (S1j − I)α j β j p = ∑ ∑ ∑ (S1j − I)α j ∑D β j p2 p1 . j∈J1 j∈J1 m∈ND 0 (18) m∈N0 Following the ideas of [3] we use the Taylor expansion 1 β j (m) = β j (ξ ) + ∇β j (ξ )T (m − ξ ) + (m − ξ )T ∇2 β j (m − ξ ) 2 (19) where ξ = ξ (t, n). Since β j is at most quadratic by Assumption 2, all higher-order terms vanish. This yields2 ∑ β j (m)p2 (t, m|n) = β j (ξ ) + R j (t, n) (20) m∈ND 0 with R j (t, n) = trace Cξ (t, n)∇2 β j because ∑m∈ND p2 (t, m|n) = 1 and ∑m∈ND (m − ξ (t, n))p2 (t, m|n) = 0. Substituting 0 0 this into (18) gives ∂t p1 = ∑ (S1j − I)α j β j (ξ )p1 + R = A(ξ )p1 + R (21) j∈J1 with a rest term R = R(t, n) given by R= ∑ (S1j − I)α j R j p1 . j∈J1 Comparing (21) with (9) yields ∂t p1 − ∂t w = A(ξ )p1 − A(φ )w + R and since p1 (0, ·) = w(0, ·) according to (11), we obtain kp1 (t, ·) − w(t, ·)k`1 ≤ Z t 0 A ξ (s, ·) p1 (s, ·) − A φ (s, ·) w(s, ·) 1 ds ` + Z t 0 R(s, ·) 1 ds. ` (22a) (22b) Our next goal is to derive a bound for the second term (22b). According to Assumption 4 we have sup sup |Cξ (t, n)| ≤ t∈[0,tend ] n∈Nd 0 C , ε whereas (2) yields 2 The remainder term R j is not to be mixed up with the reaction channel R j in (1). Error bound for hybrid models of two-scaled stochastic reaction systems ( 0 ∇ βj = ε γ( j)+1 11 if |b j | ≤ 1 if |b j | = 2. 2 By Assumption 2, no other cases have to be considered. Hence, it follows that sup sup |R j (s, n)| = sup sup trace Cξ (s, n)∇2 β j ≤ Cε γ( j) , (23) s∈[0,tend ] n∈Nd 0 s∈[0,tend ] n∈Nd 0 and Assumption 5 and the fact that γ( j) = 1 for all j ∈ J1 yield the estimate Z t 0 R(s, ·) 1 ds ≤ Ctε sup sup kα j (s, ·)p1 (s, ·)k 1 ≤ Cbtend ε. ` ` (24) s∈[0,t] j∈J1 If |b j | ∈ {0, 1} for all j = 1, . . . , r, then ∇2 β j = 0 and hence R(s, ·)`1 = 0 such that one can choose Cb = 0 in the special case (14). The first error term (22a) can be bounded by A ξ p1 − A φ w 1 ` 1 = ∑ (S j − I) β j ξ α j p1 − ∑ (S1j − I) β j φ α j w `1 j∈J1 j∈J1 ≤ C max β j ξ α j p1 − β j φ α j w`1 j∈J1 ≤ C max β j ξ p1 − β j φ w`1 j∈J1 (25) due to Assumption 5. Applying Lemma 1 now yields β j ξ p1 − β j φ w 1 ≤ Cε γ( j) ξ p1 − φ w 1 + ε −1 p1 − w 1 , ` ` ` and since the maximum in (25) is only taken over J1 , it follows that A ξ p1 − A φ w 1 ≤ C ε ξ p1 − φ w 1 + p1 − w 1 . ` ` ` Substituting (24) and (26) into (22a) and (22b) yields the assertion. (26) t u Lemma 3. Under the assumptions of Theorem 1 there are constants Cb ≥ 0 and C > 0 such that k(ξ p1 − φ w) (t, ·)k`1 ≤ Cb + C Zt k(ξ p1 − φ w) (s, ·)k`1 ds 0 + C ε Zt kp1 (s, ·) − w(s, ·)k`1 ds. 0 for all t ∈ [0,tend ]. The constants Cb and C do not depend on t or ε. If |b j | ∈ {0, 1} for all j = 1, . . . , r, then we can choose Cb = 0. 12 Tobias Jahnke and Vikram Sunkara Proof. With similar arguments as in the proof of Lemma 2, it can be shown that r ∂t (ξ p1 ) (t, n) = ∑ µ j ∑D j=1 + S1j α j β j p (t, n, m) m∈N0 ∑ ∑D m (S1j − 1)α j β j p (t, n, m) (27) j∈J1 m∈N 0 (see step 1 in the proof of Lemma 6 in [19] for details). For the first term on the right-hand side, (20) yields ∑ S1j α j β j p (t, n, m) = α j (n − ν j ) ∑ β j (m)p2 (t, m | n − ν j ) p1 (t, n − ν j ) m∈ND 0 m∈ND 0 = α j (n − ν j ) β j ξ (t, n − ν j ) + R j (t, n − ν j ) p1 (t, n − ν j ) = S1j α j β j (ξ ) + R j p1 (t, n). Moreover, it follows from (19) that mβ j (m)p(t, n, m) = β j (ξ )ξ + T j (t, n) p1 (t, n) ∑ m∈ND 0 with ξ = ξ (t, n) and 1 T j (t, n) =Cξ (t, n)∇β j (ξ ) + ξ R j (t, n) 2 1 + ∑ (m − ξ )(m − ξ )T (∇2 β j )(m − ξ )p2 (t, m | n). 2 m∈ND 0 Substituting into (27) yields r ∑ µ j S1j ∂t (ξ p1 ) (t, n) = α j β j (ξ )p1 (t, n) j=1 + ∑ (S1j − 1)α j β j (ξ )ξ p1 (t, n) + R(t, n) j∈J1 with defect r R(t, n) = ∑ µ j S1j R j α j p1 (t, n) + j=1 ∑ (S1j − 1)T j α j p1 (t, n). j∈J1 Comparing this with (10) shows that ∂t (ξ p1 )(t, n) = F(ξ , p1 ) + G(ξ , p1 ) + R. Error bound for hybrid models of two-scaled stochastic reaction systems 13 We will now prove that kR(t, ·)k`1 ≤ Cb with a constant Cb ≥ 0 which does not depend on ε nor on t ∈ [0,tend ]. In the special case (14), we have that ∇β j (ξ ) = 0 for all j ∈ J1 and ∇2 β j (ξ ) = 0 for all j = 1, . . . , r. This implies R j = 0 for all j, T j = 0 for all j ∈ J1 and hence R(s, ·)`1 = 0 such that one can choose Cb = 0. If (14) is not true, then according to (23) we know that |R j (s, n)| ≤ C for all j = 1, . . . , r, and with straightforward calculations and Assumption 5 we obtain the bound r ∑ µ j S1j R j α j p1 (t, ·) 1 ≤ C max kα j (·)p1 (t, ·)k`1 ≤ C. ` j=1 j=1,...,r Concerning the second term in R, Assumption 4 and (2) imply that sup sup |T j (t, n)| ≤ Cε γ( j)−1 . t∈[0,tend ] n∈Nd 0 The sum in the second term of R is only taken over j ∈ J1 such that γ( j) = 1. With Assumption 5, it thus follows that ∑ ∑ (S1j − 1)T j α j p1 (t, ·) 1 ≤ C max kα j (·)p1 (t, ·)k`1 ≤ C, ` j∈J1 m∈ND j=1,...,r 0 which proves kR(t, ·)k`1 ≤ Cb . Now the error ξ p1 − φ w can be estimated by k(ξ p1 )(t, ·) − (φ w)(t, ·)k`1 ≤ ≤ Z t 0 Z t 0 k∂t (ξ p1 )(s, ·) − ∂t (φ w)(s, ·)k`1 ds kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ds Z t + 0 (28) kG(ξ p1 )(s, ·) − G(φ w)(s, ·)k`1 ds +Ct. It follows from Assumption 5 and Lemma 1 that kG(ξ , p1 )(s, ·) − G(φ , w)(s, ·)k`1 ≤ C max α j β j (ξ )p1 − β j (φ )w (s, ·) `1 j=1,...,r ≤ C max β j (ξ )p1 − β j (φ )w (s, ·)`1 j=1,...,r ≤ C kξ p1 (s, ·) − φ w(s, ·)k`1 + C kp1 (s, ·) − w(s, ·)k`1 . ε (29) A corresponding bound has to be shown for F(ξ p1 ) − F(φ w). Assumption 5 yields kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ≤ C max α j β j (ξ )ξ p1 (s, ·) − α j β j (φ )φ w(s, ·) 1 j∈J1 ` ≤ C max β j (ξ )ξ p1 (s, ·) − β j (φ )φ w(s, ·) 1 . j∈J1 ` 14 Tobias Jahnke and Vikram Sunkara We decompose the error into three parts: kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ≤ C max β j (ξ )[ξ p1 − φ w](s, ·) 1 + β j (ξ )φ [w − p1 ](s, ·) 1 j∈J1 ` ` + φ [β j (ξ )p1 − β j (φ )w](s, ·) 1 . ` Since (2) and Assumption 4 imply that for every j ∈ J1 |b | sup sup β j ξ (t, n) ≤ Cε γ( j)−1 ε |b j | sup sup ξ (t, n) j t∈[0,tend ] n∈Nd t∈[0,tend ] n∈Nd 0 0 ≤ Cε and since supn∈Nd |φ (s, n)| ≤ 0 C ε γ( j)−1 = C, by Assumption 4, we obtain kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ≤ C[ξ p1 − φ w](s, ·) 1 ` C w(s, ·) − p1 (s, ·) 1 + max + β j (ξ )p1 − β j (φ )w (s, ·) 1 . ` j∈J1 ε ` Applying Lemma 1 to the last term yields max β j (ξ )p1 − β j (φ )w (s, ·) 1 ≤ Cεk ξ p1 − φ w k`1 +Ckp1 − wk`1 j∈J1 ` because γ( j) = 1 for j ∈ J1 . 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